Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds
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The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs G, H, and lists L(v)⊆ V(H) for every v∈ V(G), a list homomorphism is a function f:V(G)→ V(H) that preserves the edges (i.e., uv∈ E(G) implies f(u)f(v)∈ E(H)) and respects the lists (i.e., f(v)∈ L(v)). Standard techniques show that if G is given with a tree decomposition of width t, then the number of list homomorphisms can be counted in time |V(H)|^t· n^𝒪(1). Our main result is determining, for every fixed graph H, how much the base |V(H)| in the running time can be improved. For a connected graph H we define irr(H) the following way: if H has a loop or is nonbipartite, then irr(H) is the maximum size of a set S⊆ V(H) where any two vertices have different neighborhoods; if H is bipartite, then irr(H) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected H, we define irr(H) as the maximum of irr(C) over every connected component C of H. We show that, for every fixed graph H, the number of list homomorphisms from (G,L) to H * can be counted in time irr(H)^t· n^𝒪(1) if a tree decomposition of G having width at most t is given in the input, and * cannot be counted in time (irr(H)-ϵ)^t· n^𝒪(1) for any ϵ>0, even if a tree decomposition of G having width at most t is given in the input, unless the #SETH fails. Thereby we give a precise and complete complexity classification featuring matching upper and lower bounds for all target graphs with or without loops.
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